If you know how to multiply 2×2 matrices, and know about complex numbers, then you'll enjoy this connection. Any complex number (a+bi) can be represented by a real 2×2 matrix in the following way! Let the 2×2 matrix

[ a b ]

[ -b a ]

correspond to (a+bi). Addition of complex numbers then corresponds to addition of the corresponding 2×2 matrices. So does multiplication! Observe if you take this product:

[ a b ] [ c d ]

[ -b a ] [ -d c ]

you get

[ (ac-bd) (ad+bc) ]

[ -(ad+bc) (ac-bd) ]

which is precisely what you would get if you multiplied (a+bi) and (c+di) and then converted to a 2×2 matrix!

**Presentation Suggestions:**

Let students do the multiplication, or maybe have done it already for homework before you present this fun fact. As a follow up Fun Fact, note that taking determinants of these matrices produce Products of Sums of Two Squares.

**The Math Behind the Fact:**

The reason this works is because complex multiplication can be viewed as a linear transformation on the 2-dimensional plane. In linear algebra, you learn that every linear transformation can be represented as matrix multiplication by a suitable matrix.

**How to Cite this Page:**

Su, Francis E., et al. “Really Complex Matrices.” *Math Fun Facts*. <https://www.math.hmc.edu/funfacts>.

**Fun Fact suggested by**:

Francis Su